Negative Index of Refraction

$Negative Index of Refraction$ Negative Index of Refraction Girish Gupta School of Physics and Astronomy University of Manchester BSc Dissertation October-November 2008 Abstract Victor Veselago brought about current thinking about the consequences of both the permeability and permittivity of a material being less than zero in 1968, long before it was practically realised. Since then scientists and engineers have worked hard to bring the Russian scientist’s ideas to non-theoretical use. Most notably in the field, David Smith and John Pendry have begun work in bringing about the reality of a ‘perfect’ lens. This report aims to give the reader an idea of the theories behind the concept and of the work conducted by scientists and engineers in the latter half of the last century and the beginnings of this, on the realisation of these, specifically focusing on the perfect lens. 1. Introduction and Theory The index of refraction, n, is an incredibly fundamental concept in the fields of electromagnetism and (arguably) its offshoot, optics. Its most simple definition is as a measure of the speed of electromagnetic radiation in a material (v) compared with the speed of light ( c) in a vacuum, ͢ = . A more useful definition, however, relates n to 1 the relative permeability ( εr) and permittivity ( µr) of a material, ͢ = ±ǭ -- Equation 1 – The index of refraction defined in terms of the relative permeability ( εr) and permittivity (µr) of a material. In this paper we will be taking the negative square root of this equation. This definition of n is more fundamental as the speed of light is defined alongside the vacuum—rather than ͯͥ relative—forms of these two quantities itself, ͗ = √ . The first major paper published on the topic was in 1968 by Victor Veselago i. Back then, there was no evidence that materials with negative refractive indices were naturally occurring or could be artificially produced, yet, Veselago wrote his 1968 paper on a completely theoretical basis, hoping that its rigorously-argued conclusions would one day become viable. Veselago was not the first scientist to hypothesise about the implications of a negative index of refraction ii . In 1904, Horace Lamb suggested the idea of “backward waves” in reference to pressure waves iii . In the same year, Arthur Schuster went on to quote Lamb’s work and extend the idea to electromagnetic waves in his book Theory of Optics and drew up a diagram similar to Figure 2. However, the mention of negative index of refraction by Schuster was a minimal ending to a chapter entitled ‘Transmission of Energy’ iv . It was Veselago’s work in 1968 that delved deepest into the repercussions of negative index materials 1,v. In the paper, the Russian noted that rather than taking the positive square root of Equation 1, he would like to work through the implications of taking the negative root. He admitted that there could be three possibilities arising from doing this: 1 Materials with negative refractive index were called ‘left-handed’ by Veselago, the reason for which is explained in the caption for Equation 3. John Pendry and David Smith v decided to go with the name ‘negative index materials’ so as to not confuse this phenomenon with chirality. This is the name that this paper will continue to use. - 2 - • No difference will be made to a material’s properties. The negative root will have the same physical effect as the positive root. • The negative root is physically impossible. • “It could be admitted that substances [such as this] have some properties different from those … with positive ε and µ.” It was the third possibility that spurred Veselago on to continue his research— and eventually the one that was realised to be the case. He worked through Maxwell’s equations of electromagnetism and was soon forced to bring in the wave propagation vector k. The wave vector, as it is commonly called, represents the direction of wave propagation, similarly—but not identically—to the Poynting vector, S. The vector form of the wave vector is not always required and its mathematics is quite useful in scalar form, k, the wave number. The wave number is defined as, ! ͟ = ͢ ͗ Equation 2 – The wave number is important in electromagnetism. Its vector part appears to point in the wrong direction when the index of refraction is taken to be negative. where n is, again, the material’s refractive index and ω is the angular frequency of the radiation. This quantity is a major player in the propagation of electromagnetic waves. It can be found in all fundamental equations of the topic so when, as Veselago realised, its sign is reversed as that negative square root (of Equation 1) is taken, some interesting things will happen. It is worth now bringing in a relationship between the magnetic (B) and electric (E) fields that, quite evidently, pervade electromagnetism. The relationship neatly incorporates our wave vector, k, 1 ̈ = ̫ × ̋ ͗ Equation 3 – It was an equation similar to this that spurred Veselago into describing his hypothesised materials as ‘left-handed’. The equation, when k is negative, forms a left-handed set of vectors. - 3 - The relationship between the three most fundamental vectors in electromagnetism can now be seen 2. As can be seen by Equation 2, changing the sign of the refractive index will do the same to the wave number and vector. It can be seen by Equation 3 that if k is positive, B, k, and E will form a right-handed set of vectors. However, if is it negative, a left-handed set will be formed. 3 The Poynting vector was mentioned earlier as being similar to the wave vector. The vector describes the direction in which energy is carried by the wave. It is defined as, 1 ̙ = ̋ × ̈ Equation 4 – The Poynting vector points in the direction of energy flux. Surprisingly, this appears to be negative too when the material has a negative index of refraction. Again, the three vectors in Equation 4 will form a right-handed set with each other unless µ is negative, in which case you will again get a left-handed set of vectors. Here is where things begin to get interesting. If the direction of the Poynting vector is reversed, the implication is that the energy is flowing the ‘wrong’ way. The energy flow represented by S is in the opposite direction to k. It is easier to see this by bringing in the wave velocity, ! ̶ = ̰ ̫ Equation 5 – The phase velocity appears to be negative. Veselago’s genius was in noting that there are different forms of velocity so the idea is not as paradoxical as it sounds. It is clearly seen by Equation 5 that the wave velocity and wave vector will always point in the same direction (assuming the angular frequency is positive). With a negative wave vector, this brings about a startling effect—the velocity of the wave will appear to go backwards, from receiver to source. 2 In his 1968 paper, Veselago actually does the maths with the magnetic field strength vector, H. However, there is no need to introduce this quantity here so the algebra has been worked through to remove it. 3 Veselago went on to introduce a matrix of which he called the determinant the ‘rightness’ of a material, p. If the set of E, B, and k was positive, p would equal +1; if the set were negative, p would equal -1. - 4 - Veselago’s genius was in noting that there are many different methods of defining the velocity of a wave. Equation 5 is in fact the phase velocity of the wave, hence the subscript p. This is the most common form of the velocity and simply denotes the rate at which any point on the wave (a peak or a trough, for example) passes a point in space. However, as waves become more complex than a coherent, monochromatic beam, which is, of course, the case in nature, other velocities need to be brought in. The fact that dispersion occurs, i.e., the refractive index being a function of wavelength, is also important. The group velocity is slightly different and brings in more realism. It takes into account the envelope that carries the wave that the phase velocity defines. It is defined by, ͘! ̶ = ̧ ̫͘ Equation 6 – The group velocity describes the velocity of the wave packet. If the wavelength is constant, the group velocity is equal to the phase velocity. This group velocity will not have the same sign as the phase velocity. The Poynting vector will by definition have opposite sign and the rest of this report will work through the implications of the fact that the energy flux (direction given by the Poynting vector) has an opposite sign to the phase velocity. 2. Physical Consequences There are a number of consequences of the above theories. Any equation featuring those variables we are interested in making negative should be of interest to us. Fortunately, those variables, µ, ε, n and their derivatives, pop up all over electromagnetism and optics. I will describe the theory behind some of the physical consequences and go on to describe how they have been investigated since Veselago’s 1968 paper. Doppler Shift Doppler shift is a well known phenomenon that occurs for both electromagnetic and pressure (sound) waves. Though we are dealing with electromagnetism, the more obvious and ubiquitous example is of the emergency vehicle with sirens wailing - 5 - coming towards and then shooting off passed, the observer. As the vehicle’s sirens approach, the frequency of the sound increases.

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